On Solutions in Differential Algebraic Closure, Finite Representation of Transcendental Functions, and Their Application Framework to Number Theory Problems: A Unified Spectrum from Quadratic, Cubic, Quartic to Equations of Arbitrary Degree

This paper systematically constructs a unified solution method for polynomial equations based on differential algebraic closure and proves its strict equivalence to classical transcendental function solutions.The core contributions lie in: (1) Constructing the universal differential algebraic closure Kn and its specialization homomorphism, providing explicit finite closed-form solutions for polynomials of any degree n; (2) Proving explicit isomorphisms between these differential algebraic solutions and solutions expressed via trigonometric, hyperbolic, elliptic, hyperelliptic, and Riemann θ functions, thereby building a bridge between finite operations (arithmetic, differentiation, evaluation, radicals) and transcendental functions;(3) Conversely demonstrating that a large class of transcendental functions (solutions of algebraic differential equations) can be finitely represented by differential algebraic solutions of polynomial equations, thus precisely characterizing the class of “finitely representable functions” and clarifying its relativity—dependent on the chosen set of basic operations and constants; (4) Based on this unified framework, proposing innovative, potentially computable differential algebraic research pathways for cutting-edge number theory problems (including the Birch–Swinnerton-Dyer (BSD) conjecture, class number problems, the abc conjecture, and an explicit formulation of the Langlands program). This work not only expands the boundaries of the Abel–Ruffini theorem by introducing and characterizing the differential algebraic complexity class DA, but also unifies the classical spectrum of polynomial equation solving—from quadratic, cubic, quartic to quintic and beyond—within an explicit, coefficient-driven finite representation framework, offering potential constructive proof tools for several outstanding problems in number theory and arithmetic geometry.